Abstract. Dirk Huylebrouck and Patrick Labarque try to provide
a positive answer to the question that the golden section corresponds
to an optimal solution. It is but a college-level rephrasing exercise,
but it could reboot the mathematical career of the golden section.
An extension to the related silver section is given as well. The
authors betin their examination with the definition of the golden
number, then proceed to its applications to architecture, grey-tone
mixing, colour mixing and bicycle gears.

INTRODUTIONThe metallic means are the positive roots
of x2 - nx -1
= 0, for different values of the positive integer n. The
first (n =1) is the most prominent member, the golden
mean f, while for n = 2 the
quadratic yields sAg
the silver mean and the sBr bronze mean for n = 3 .
Literature describes their properties extensively, often including
wild statements of doubtful verification, repeated over and over,
based on identical erroneous sources. Some false applications
are simply results of a pseudoscientific new age imagination,
while others build on approximate measurements allowing any desired
interpretation. The so-called omnipresence of the mythical divine
proportion should be assessed critically.

Nevertheless, colour theory already forced the authors to
recognise a new true application of the metallic means. A partitive
mixing of 2 complementary colours is "well-balanced and
pure" if rotation of a disk covered for 50% each by these
colours yields a 50% grey. Yet, the colour or grey promised by
the software is rarely the one of the final output. It turns
out that proportions of grey tones related to the metallic means
play interesting roles in partitive mixing. This will be described
in Section 2, and Section 3 will show extensions of these results
to "golden colours", whose Red, Green, and Blue components
are each of the proportion 1/f. Some
related computational recreations illustrate this colourful theory.

More recently, the authors were again surprised to come across
the golden number, in the completely different field of the determination
of gears and the addition of speeds. This will be described in
Section 4. Together with the colour theory applications, it shows
that the appearance of the golden number is not always the result
of a far-fetched imagination, like many statistical and historical
'studies' about the golden number. For instance, it has often
been repeated, with dubious justification, that the rectangle
of width 1 and length is considered as the most elegant one.
Yet, in Section 5 we will answer a question concerning the optimality
of the golden rectangle in artistic paintings or buildings' facades.
This could reboot the mathematical career of the metallic means.

1. THE GOLDEN NUMBERThe golden number (or golden
section, mean, or ratio, or divine proportion, etc.) arises when
a line segment of length x (>1) is divided into two
pieces of lengths 1 and x-1, such that the whole length
is to 1, as 1 is to the remaining piece, x-1. Thus, the
x / 1 ratio must equal the ratio 1/(x-1). This
produces the equation x2
- x -1 = 0, of which f = (1+Ö5)/2 »1.6180
is the positive solution; note that f-1=
1/f»0.6180.
More generally, the positive roots of x2
- nx -1 = 0 yield the family of metallic means, for n=1,
2, .... For n=2, the positive root is the silver mean
sAg
= 1+Ö2; for n=3, it is
the bronze mean sBr
= (3+Ö13)/2, etc.

Literature often describes the golden section as a cookbook-like
formula. The historical justification is but a recipe: the (unequal)
division of a line such that the whole length stands to the larger
piece, as the larger is to the smaller. The authority of Euclid
may be a good reason for using this recipe, which was termed
simply "division in extreme and mean ratio". What importance
Greek mathematicians attached to this number is unknown, and
after all, the adjective "golden" dates from only 1844
[Markowsky 1992: 4]. The properties of these numbers have been
described extensively (for a comprehensive survey, see [Spinadel
1998]). Of course, half the diagonal of a rectangle with length
2 and width 1 corresponds to the 'irrational part', Ö5/2,
of the golden section, but this is so obvious that it is hardly
a "mathematical application". The only related, straightforward
yet surprising geometric properties are perhaps those of the
pentagon. Therefore, the related angle of 36° is perhaps
appropriately called the "golden angle" (Figure
1).

Some authors contend that the golden number is "the most
irrational of all irrational numbers", with the other metallic
means following in order of decreasing irrationality, because
their representations in continued fraction form are the slowest
converging. However, this justification does not use the conventional
mathematical approach, as it does not specify when a number is
more or less irrational than another number. To understand this,
note that since x2 -
x -1 = 0 can be rewritten x=1+1/(1+1/x),
replacement of x by its expression in the denominator
yields x=1+1/(1+x), which, when repeated ad infinitum,
yields the continued fraction form for the solution,

,

and similarly,

Continued fractions converge more quickly when the successive
denominators are large, but since the continued fraction expansion
for f always involves adding only
a 1 (the smallest possible positive integer) to the denominator,
it converges the slowest (cf. [Spinadel 1998: 3-8]). While this
may be important in some contexts, it is by no means standard
to judge the "irrationality" of a number by the rate
of convergence of its continued fraction expansion.

The lack of straightforward mathematical reasons to justify
a prominent status of the golden section forces some to evoke
aesthetical, historical, or natural considerations. The golden
rectangle of width 1 and length f
would be the most elegant one, as various designs would show.
Yet, strong arguments show that the Greeks did not use it deliberately
in their Parthenon (Figure 2), nor was the Great Pyramid necessarily
designed to conform to f. Certainly
Herodotus did not mention such a description; it simply does
not appear in his text, as anyone can check. Nor does Leonardo's
early work (nor the UN building, nor the human body, nor Virgil's
Aeneid) constitute evidence of the golden ratio's exceptionality
[Markowsky 1992]. Finally, the often-cited association of the
golden rectangle spiral to that of the nautilus shell is seen
to be incorrect upon a more thorough analysis [Sharp 2002].

Still, the Fibonacci sequence

1, 1, 2=1+1, 3=2+1, 5=3+2, 8=5+3, 13=8+5,

is linked to f, since the quotient
sequence, formed by taking the quotient of each term divided
by the previous one, tends to f. Painters
and sculptors often subdivide their (rectangular) canvas or stone
using specific proportions, and 3 : 5 or 5 : 8 arrangements are
frequent. This provides a connection to the Fibonacci numbers.
Yet, 2 : 4 : 8 schemes are popular too and even used by great
masters such as Seurat, Monet and Cézanne. It would be
hard to prove that Fibonacci subdivisions lead to the "most
beautiful" artworks. Furthermore, some contradiction is
involved in the statement that golden section subdivisions are
the key to fine art creations - why then would artists and their
creative skills still be needed instead of a golden section computer
program?

In the last resort, if no mathematical, aesthetical or historical
reasons help, some turn to statistics. Yet, Markowsky asserts
that the statistical experiments about the golden rectangle do
not seem to confirm a preference for the golden rectangle. It
is not indisputable that one can pick out the "most elegant"
(golden) rectangle from an arbitrary list. Other examples seem
mere repetitions of the same unreliable statistical studies.
Matila Ghyka was one of the prominent advocates of the golden
section myth, and his popular books influenced many readers,
such as Le Corbusier. Of course, there is nothing wrong with
the fact that architects use mathematics or arithmetical proportions
as inspiration, instead of some romantic, ecological, ergonomic
or other kind of muse to create their works of art. It even puts
mathematics in a position of great honour that someone like Le
Corbusier uses it. Nevertheless, these claims are artistic ones,
without scientific consequences.

In spite of the vigorous considerations in disfavour of the
golden section, many mathematicians continue to spread the myth.
Mathematicians may be somewhat biased about the subject, maybe
because they appreciate the attentions artists want to pay them,
for once. A moderate conclusion comes from a psychologist, Christopher
D. Green:

Although many researchers have concluded that the effect
is illusory, more carefully conducted studies have fairly consistently
shown that there is, in fact, a set of phenomena that require
explanation, though no one has yet produced an explanation, both
adequate and plausible, that has been able to stand the test
of time. [Green 1995]

Nevertheless, we believe two cases can be added to the (not
so lengthy) list of true applications of the golden section.
The first is about colour theory, while the second stems from
still another field, i.e., the study of planetary gears. In addition,
the last section below concerns a classical mathematical justification
for the interpretation of the golden section as an optimal solution,
despite the previous critical considerations. Thus, maybe there
is something more to be discovered about that (in)famous golden
section.

2. MIXING GREY TONESIn much computer-graphics software, the
user can numerically determine the kind of grey to use in shading
a surface. Actually, the W-value of a grey surface indicates
the amount of white in it (in most software, the amount of black
is used instead). Thus, a W-value of 0 corresponds to black,
while a W-value of 1 means in fact white, or, more precisely,
"maximum light". Unfortunately, the value given by
the software does not necessarily guarantee an identical result
by the hardware. In the graphical sector, the determination of
the exact grey value of a drawing is important, and some designers
apply several transparent layers on top of each other on a white
background to study this problem. Obtaining white through two
layers corresponds to having white on the first AND on the second.
Thus, appealing to probability theory, if W1
is the W-value of the first layer, and W2
the value of the second, their product provides the resulting
W-value of the double layer. (Given two independent events A
and B, the probability of both A and B occurring
is given by their product: Pr(A and B)=Pr(A)Pr(B).)
The illustration shows such a subtractive mixture (see the upper
portion of Figure 3).

Rapidly rotating a disk with two different grey tones is another
technique to judge the final impression of a black and white
mixture. At each moment, a pixel from one grey tone OR from the
other grey tone appears. Thus, the probability of seeing one
of the two pixels in the rotating result will correspond to their
weighted average, i.e., if the W1
grey covers a r1
fraction of the disk, and the W2
grey a r2
fraction of the disk (r1+r2=1), the rotation will show a grey with
value r1W1+r2W2.
Designers call this procedure a partitive mixture. Often,
the two grey tones cover the disks in sectors, since their fractions
can thus be more easily measured (see the lower portion of Figure 3).

These definitions are not of mere theoretical importance.
Partitive mixture, as a reference, can be used to determine the
W value of a transparent layer. For the study of colour mixing
in the next section, combinations of transparent coverings of
a single and a double layer are used. In preparation to this,
we first work some similar operations with grey tones. On a rotating
disk, suppose a single layer of W grey covers a (1-r)
fraction of the disk, and a double layer of W grey covers a r fraction. Then their mixture corresponds
to a grey tone with value w given by (1-r)W+rW2=w.
This produces the quadratic equation W2-(1-1/r)W-w/r=0.

The following natural question arises: given a single layer
covering a given share (fraction) of a disk, is there a corresponding
grey value W such that the grey resulting from the rotation
has a w value equal to that share (fraction)? Thus for
any positive integer n, w=r=1/n
provides a family of interesting particular cases. It transforms
the quadratic into

W2-(1-n)W-1=0.

For n=2, the equation is W2+W-1=0,
and its solutions are -(1±Ö5)/2,
or approximately 0.618 and -1.618.

The first of these values, 0.618, is the inverse of the golden
number and the second, -1.618, its opposite. Rejecting the negative
value, an approximate 61.8% grey provides the solution. Summarising,
if a disk is half covered by a 61.8% grey and the other half
by a double layer of this grey, then the rotation yields a resulting
grey that is 1/2 white and 1/2 black. Similarly, if a disk is
2/3 covered by a Ö2-1»41.4%
white grey and the other 1/3 by a double layer of this grey,
then the rotation yields a 1/3 white result. For 3/4 coverage
by a (Ö13-2)/2 »30.3%
grey and the other 1/4 by a double layer of this grey,
then the rotation yields a 1/4 white result.

3. COLOUR MIXING
The reason for covering the disk by another double layer stems
from colour theory. Here, subtractive and partitive mixing of
two (or more) colours are defined similarly, and if a disk is
divided in two sectors, one sector by a given colour, and the
other by its complementary colour, rotation can yield grey for
adequate proportions. That complement can correspond to a double
layer of two different colours, and if this is the case when
exchanging all three colours two by two, they are called well
balanced.

In particular, suppose that the three colours are tones of
the colours Cyan, Magenta and Yellow (the additive primary colours).
Their RGB values refer to the amounts of Red, Green and Blue
(the subtractive primary colours) light in them, as was previously
done for the amount of white. That is, a colour with an R-value
of .30 has 30% of Red light. If all RGB values are 0, black is
obtained, while 3 values of 1 yield "maximum white light".
Three arbitrary but identical RGB values yield grey. The RGB
values of Cyan, Magenta and Yellow can be placed in vectors:

C=(Rc,
Gc, Bc), M=(Rm, Gm,
Bm) and Y=(Ry, Gy,
By).

Firstly, we compare the Red value Rc
of Cyan to those of its complementary colour obtained by the
subtractive mixture of Magenta and Yellow. The white percentage
of the grey obtained by spinning the disk is unknown and will
be denoted w1. If Cyan
covers a (1-r1
) fraction of the disk, the grey that is obtained corresponds
to

(1-r1)Rc+r1RmRy=w1.

Proceeding similarly for the Red components of Magenta and
Yellow produces a system of 9 equations, and 9 variables. The
equations of the system can be solved in groups of three, comparing
the amount of R, G and B separately. For
the R coordinates, for instance, it is enough to study
the system:

(1-r1)Rc+r1RmRy=w1(1-r2)Rm+r2RyRc=w2(1-r3)Ry+r3RcRm=w3

This yields a fifth-degree equation in Rc, which, in general, would require
the use of elliptic functions to solve. However, this complicated
method is not necessary in the special cases we will consider,
since an easy factorisation of the equation will be possible.
Indeed, in colour theory, complementary colours are called "well-balanced
and pure" when their mixture yields a neutral 50% grey or,
using our rotating disks w1=w2=w3=0.5
and r1=r2=r3=0.5.
In this ideal case, the wiand
ri
values are identical. More generally, we could look for the Red-Green-Blue
combination that provides identical grey tones when the rotating
disks are also divided in the same way. Indeed, we want to look
for those Red-Green-Blue combinations, which provide a grey with
a given % white, if the rotating disks are all divided identically.
Thus, w1=w2=w3=s
and r1=r2=r3=t.
This special case causes the fifth-degree equation to have the
factorisation:

(tRc+t-1)2 ((t2-t)Rc+2t+st-t2-1)(-tRc2+(t-1)Rc+s)=0

The third factor corresponds to the quadratic equation

Rc2+(1/t-1)Rc-s/t=0

Thus, if s=t and t=1/(1+n), the
factor corresponds to Rc2+nRc-1=0.
For positive integers n, its solutions are the inverses
of the metallic means. Setting X=1/Rc,
it reduces to the standard form X2-nX-1=0,
the positive solutions of which are the golden number for n=1,
the silver for n=2, and the bronze for n=3. Combination
of the solutions (0,1,1), (1,0,1), (1,1,0) and (1/f,
1/f, 1/f)
for each component yields a set of golden colours. Thus, if a
rotating disk is covered for 1/2 of one colour and the other
half of its complement obtained by mixing the other two colours,
then a 1/2 white and 1/2 black result is obtained, if 1/f RGB values are used (or the trivial 0
and 1 values).

The proposed mathematical tools have interesting or at least
amusing applications on the well-known problem of getting the
same colour in a print-out and on screen. Other applications
are the question of finding the subdivision to be used as an
approximation of a given colour using a pattern (partitive mixture)
of a set of given colours, in discrete amounts, such that there
is no combination that yields the desired colour (see [Huylebrouck
and Labarque 2001]; Figure 4).

4. BICYCLE GEARSAnother new and true application of the
golden section was found in a special system for bicycle gears
that uses planetary wheels enclosed inside a gear cage. The basic
3-speed hubs have a single "sun" wheel. A ring with
3 identical "planet" wheels meshes with and revolves
around the sun wheel. The planet wheels are surrounded by an
inside out cog wheel (Figure 5).

We indicate the central sun wheel by the letter A. Supposing
its radius is 1, the length of the arc on its circumference corresponds
to the change of the angle a (in radians)
in its centre. The middle ring with three planetary wheels on
it is represented by B, and its angular change by b.
Finally, g will be the angular change
of the outside wheel C, with radius x (x³1).
Any of the three wheels A, B and C can serve as drive wheels,
while one of the two remaining wheels can be fixed to the frame.
The last remaining wheel then provides the desired result.

For instance, suppose that wheel A is fixed, B the drive wheel,
and C the wheel that yields the result. The resulting arc gx on C is composed of two arcs,
one represented in a dashed thick line, bx,
and another given in a thick uninterrupted line. The length of
the latter arc is drawn on one planetary wheel as well (twice;
thick lines), and can thus be reconstructed on fixed wheel A,
where its value equals b×1.
An addition of both terms yields the comparison: gx=b+bx.
Thus, the resulting gear is g/b=(1+x)/x.

A comparison to the other gears yielded the ranking given
in Table 1,
in increasing order. Some numerical values were given too (up
to three decimals). The forward gears (gears 3-7) form a geometric
progression if and only if gear7/gear6 = gear6/gear5 = gear5/gear4
= gear4/gear3, or x2=1+x.
This is the quadratic equation of which the golden number provides
the positive solution. That is, a (positive) radius x=f is such that each gear is the previous
one multiplied by f»1.618, in
a planetary gear system. Vice versa, it is the only case where
such a sequence of gears consecutively increases by a constant
factor, which happens to be f. Still,
cyclists will prefer other types of bicycle gears, for several
other practical or physiological reasons.

5. OPTIMAL SOLUTIONSThe new true applications about golden
colours and bicycle gears motivated the authors to have another
look at that (in)famous golden rectangle. Inspired by completely
different mathematical problems (see [Huylebrouck 2001]), a common
mathematical set-up for an optimisation problem is proposed.
A function that describes a certain geometric construction is
defined and its derivative is set equal to zero, in order to
find exceptional solutions, and the golden section is one of
them.

The geometric construction concerns a rectangle of arbitrary
length a and arbitrary width b. We increase a
by x and b by y to create two rectangles
on the main diagonal (in grey), and two rectangles on the border
(in white). The extended area, made up by the rectangles on the
border and the added grey rectangle, is bx+ay+xy
(surrounded be a thick black line). We compare this (as a ratio)
to area ab+xy on the main diagonal (shaded) as
in Figure 6.

This produces a function:

.

Its extreme value is computed by setting both the partial
derivative with respect to x and with respect to y
equal to 0:

, or
.

Solving these equations simultaneously yields the positive
solution

,

corresponding to a saddle point. The z-value is the
golden number, independently of the value of a and b.
(This could have been expected, as an initial division by a
and by b would have simply rescaled the problem.) The saddle
point corresponds to maximum provided that x=y,
and this leads to the special case of extending a square by another
square to form a larger square: the added area is maximal, with
respect to the area of the two squares, if the two non-square
rectangles are golden rectangles (Figure
7).

The silver section can be obtained through such an optimisation
procedure, but we summarise it in a somewhat simplified construction,
starting with a square (a=b=1 ). Now, the area
cut off at one side is compared to the total area of the two
squares. The function to be maximised is therefore

The given formulation corresponds to what is most probably
intended by the expression of "optimal solution", as
given by so many authors, and refuted by others. It proposes
a rectangle that is neither too small nor too large, with respect
to a given verifiable objective criterion. It seems but a rephrasing
exercise, using mathematical vocabulary, but it could help to
reboot the mathematical career of the golden section.

It is another matter to state that the given criterion produces
the most elegant rectangle. Nevertheless, psychological golden
section tests may be reformulated as "finding the rectangle
such that the added area is maximal when compared to two squares
constructed on the diagonal". People already have difficulties
in choosing the right shape of a rectangle simply because estimating
sizes is not easy. This obstacle may be important when making
judgements about elegance of shapes and sizes.